Optimal linear discriminators for the discrete choice model in growing dimensions

Asymptotic properties of the maximum smoothed partial likelihood estimator in the change‐plane Cox model

“…Utilizing a maximal inequality of Giné and Guillou (2001), our close‐to‐$$$ {n}^{-1} $$$ rate of the classification parameter is faster than most of the rates of the smoothed estimators in the change‐plane model literature that are shown to be at most $$$ {n}^{-3/4} $$$ (e.g., Li et al, 2021; Seo & Linton, 2007a; Zhang et al, 2021). Furthermore, although Mukherjee et al (2020) derives the same close‐to‐$$$ {n}^{-1} $$$ rate as we do in their model, our proof is original in that we directly deal with the score function when deriving the convergence rate. Without invoking the general rate theorem for M‐estimator (e.g., theorem 3.4.1 of van der Vaart & Wellner, 1996) as Mukherjee et al (2020) do, our direct approach shortens the proof.…”

“…Furthermore, although Mukherjee et al (2020) derives the same close‐to‐$$$ {n}^{-1} $$$ rate as we do in their model, our proof is original in that we directly deal with the score function when deriving the convergence rate. Without invoking the general rate theorem for M‐estimator (e.g., theorem 3.4.1 of van der Vaart & Wellner, 1996) as Mukherjee et al (2020) do, our direct approach shortens the proof.…”

“…For example, see assumption 3(e) in Seo and Linton (2007a) and condition 6 in Li et al (2021). However, for a binary classification model, similar to this study, Mukherjee et al (2020) assume $$$ \left(\log n\right)/\left(n{h}_n\right)\to 0 $$$. Assumption 6 is similar to assumption 2(e) of Seo and Linton (2007a), though our assumption is more restrictive in assuming the boundedness of the first derivative.…”

“…Assumption 3(b) is for identifiability and is usually assumed in the change‐plane models. For example, see Assumption 1(d) of Seo and Linton (2007a) and Assumption 1 of Mukherjee et al (2020). This assumption can be weakened to allow for bounded $$$ V $$$ as “Conditionally on $$$ X $$$, the distribution of $$$ V $$$ has a positive density on an interval $$$ A $$$ such that $$$ \operatorname{inf}A<{\operatorname{inf}}_{\psi \in {\Theta}^{\psi },x\in \operatorname{supp}(X)}-\mid {x}^{\prime}\psi \mid $$$ and $$$ \sup A>{\sup}_{\psi \in {\Theta}^{\psi },x\in \operatorname{supp}(X)}\mid {x}^{\prime}\psi \mid $$$,” where $$$ \operatorname{supp}(X) $$$ is a support of the distribution of $$$ X $$$.…”

“…To achieve such flexibility, the literature has developed several versions of the change‐plane model. For example, refer to Seo and Linton (2007a), Yu and Fan (2021), Lee et al (2021), and Li et al (2021) for linear regression; Mukherjee et al (2020) for binary classification; and Zhang et al (2021) for quantile regression.…”